Question
The graphs show the percentage of high school seniors who had ever used alcohol or marijuana. The data can be described by the following mathematical models, where n is the number of years after 1990, A is the percentage of seniors using alcohol, and M is the percentage of seniors using marijuana.
Aequalsminus0.7nplus57
Mequalsminus0.4nplus21
Aequalsminus0.7nplus57
Mequalsminus0.4nplus21
Answers
GPT-4o mini
The mathematical models you provided describe the percentage of high school seniors who had used alcohol (A) and marijuana (M) based on the number of years after 1990 (n). Here's a breakdown of each equation:
1. **Alcohol Usage (A)**:
\[
A = -0.7n + 57
\]
- This equation indicates that for each passing year after 1990, the percentage of seniors using alcohol decreases by 0.7%. The intercept (when \(n = 0\), or the year 1990) gives a starting percentage of 57%.
2. **Marijuana Usage (M)**:
\[
M = -0.4n + 21
\]
- Similarly, this equation shows that the percentage of seniors using marijuana also decreases over the years by 0.4% for each year after 1990. The starting percentage (when \(n = 0\)) for 1990 is 21%.
### To analyze further:
1. **Finding values for specific years:**
- For example, if you want to know the percentages for a specific year, calculate \(n\) by subtracting 1990 from that year.
**Example: To find percentages for 2000 (n = 10):**
\[
A = -0.7(10) + 57 = -7 + 57 = 50\%
\]
\[
M = -0.4(10) + 21 = -4 + 21 = 17\%
\]
2. **Finding when usage reaches zero:**
- Set the equations to zero:
For alcohol:
\[
0 = -0.7n + 57 \implies 0.7n = 57 \implies n = \frac{57}{0.7} \approx 81.43
\]
- This means that alcohol usage could theoretically reach zero around the year \(1990 + 81.43\) (approximately 2071).
For marijuana:
\[
0 = -0.4n + 21 \implies 0.4n = 21 \implies n = \frac{21}{0.4} \approx 52.5
\]
- This indicates marijuana usage could theoretically reach zero around the year \(1990 + 52.5\) (approximately 2042).
### Summary:
These linear models illustrate a declining trend in usage amongst high school seniors for both substances, with alcohol usage dropping faster than marijuana usage. You can use these equations to predict future trends or gauge historical usage percentages by adjusting the variable \(n\).
1. **Alcohol Usage (A)**:
\[
A = -0.7n + 57
\]
- This equation indicates that for each passing year after 1990, the percentage of seniors using alcohol decreases by 0.7%. The intercept (when \(n = 0\), or the year 1990) gives a starting percentage of 57%.
2. **Marijuana Usage (M)**:
\[
M = -0.4n + 21
\]
- Similarly, this equation shows that the percentage of seniors using marijuana also decreases over the years by 0.4% for each year after 1990. The starting percentage (when \(n = 0\)) for 1990 is 21%.
### To analyze further:
1. **Finding values for specific years:**
- For example, if you want to know the percentages for a specific year, calculate \(n\) by subtracting 1990 from that year.
**Example: To find percentages for 2000 (n = 10):**
\[
A = -0.7(10) + 57 = -7 + 57 = 50\%
\]
\[
M = -0.4(10) + 21 = -4 + 21 = 17\%
\]
2. **Finding when usage reaches zero:**
- Set the equations to zero:
For alcohol:
\[
0 = -0.7n + 57 \implies 0.7n = 57 \implies n = \frac{57}{0.7} \approx 81.43
\]
- This means that alcohol usage could theoretically reach zero around the year \(1990 + 81.43\) (approximately 2071).
For marijuana:
\[
0 = -0.4n + 21 \implies 0.4n = 21 \implies n = \frac{21}{0.4} \approx 52.5
\]
- This indicates marijuana usage could theoretically reach zero around the year \(1990 + 52.5\) (approximately 2042).
### Summary:
These linear models illustrate a declining trend in usage amongst high school seniors for both substances, with alcohol usage dropping faster than marijuana usage. You can use these equations to predict future trends or gauge historical usage percentages by adjusting the variable \(n\).